In this section, we scrutinize the issue of gauge dependence in our conclusions drawn above.
It has been pointed out that the SM one-loop effective potential depends on the gauge parameter \(\xi \)  in the \(R_{\xi }\)  gauge
due to the thermal corrections to the masses.
Ref. [1]} proposed a method to determine the gauge-independent \(v_c\)  and \(v_c/T_c\)  in the high-temperature approximation.
With this method, a gauge-invariant, perturbative computation of \(v_c\)  and \(v_c/T_c\)  can be made by retaining only the quadratic
temperature-dependent terms in the effective potential (see Refs. [2]}, [3]}, [4]}, [5]}, [6]} for a systematic treatment of the high-temperature effective theory for a number
of SM extensions).
